NONLINEAR WAVES IN NEWTON'S CRADLE AND THE DISCRETE p-SCHRÖDINGER EQUATION

Abstract

We study nonlinear waves in Newton's cradle, a classical mechanical system consisting of a chain of beads attached to linear pendula and interacting nonlinearly via Hertz's contact forces. We formally derive a spatially discrete modulation equation, for small amplitude nonlinear waves consisting of slow modulations of time-periodic linear oscillations. The fully nonlinear and unilateral interactions between beads yield a nonstandard modulation equation that we call the discrete p-Schrödinger (DpS) equation. It consists of a spatial discretization of a generalized Schrödinger equation with p-Laplacian, with fractional p > 2 depending on the exponent of Hertz's contact force. We show that the DpS equation admits explicit periodic traveling wave solutions, and numerically find a plethora of standing wave solutions given by the orbits of a discrete map, in particular spatially localized breather solutions. Using a modified Lyapunov-Schmidt technique, we prove the existence of exact periodic traveling waves in the chain of beads, close to the small amplitude modulated waves given by the DpS equation. Using numerical simulations, we show that the DpS equation captures several other important features of the dynamics in the weakly nonlinear regime, namely modulational instabilities, the existence of static and traveling breathers, and repulsive or attractive interactions of these localized structures.

Breather Solutions of the Discrete p-Schrödinger Equation

“…with finite power). Moreover, static breather solutions decay doubly-exponentially in space, as numerically illustrated in [12] (see also [14] for an analytical decay estimate).…”

“…The DpS equation was recently introduced to describe small amplitude oscillations in a class of mechanical systems consisting of a chain of touching beads confined in smooth local potentials [12,13,26], the most well known example of such systems being Newton's cradle [10]. In this context, the p-Laplacian involved in (1) accounts for the fully-nonlinear character of Hertzian interactions between beads (with p = 5/2 in the case of contacting spheres).…”

“…pinned to some lattice sites and time-periodic) or traveling along the lattice. Such solutions can be generated from localized initial conditions or may arise from modulational instabilities of periodic waves [12,13,26]. Although such properties are classical in the context of anharmonic Hamiltonian lattices [9], energy localization is particularly strong in the DpS equation.…”

“…The case i = 1 yields the so-called even-parity modes (or site-centered solutions), and the case i = 2 corresponds to odd-parity modes (bond-centered solutions). These solutions were numerically computed in [12,26,13] for p = 5/2 (see also [9,23] and references therein in the case p = 4 with an additional on-site cubic nonlinearity).…”

“…The corresponding variables will be denoted by force variables and mixed (amplitude-force) variables, in analogy with the Newton's cradle system [12]. For spatially homogeneous systems, the force variables described in section 2.1 allow to convert the stationary DpS equation into a generalized DNLS equation with an on-site (fractional power) nonlinearity, which allows to prove the existence of homoclinic orbits by classical arguments (see section 4).…”